If there had been DNA tests, or even blood tests (assuming the father was known), the event wouldn’t have happened.
In the case of a baby mix-up (they do happen, now and then), there is no halachic issue in using scientific testing (note, this would be “maternity” testing). For other matters such as paternity testing (to determine if the child is a mamzer) there would be issues. To determine Jewishness (Yichus), other than by confirming the mother-child relationship, it would be very problematic.